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[[234,156,2]] d =
n
234
k
156
d
2
kd²/n
2.667
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 2 · witness weight 2 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 2 qubits)
[123, 131]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 2 qubits)
[90, 100]
certificate exact, d = 2 · CryptoMiniSat 5.14.7 SAT
X: no logical < 2 exists; Z: no logical < 2 exists

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 9 · H_Z 9
qubit degrees H_X 3–6 (mean 4.5) · H_Z 3–6 (mean 4.5)
trapping sets H_X (1,3)×117 (2,0)×234 (3,0)×1053 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 117 (1,6): 117 (2,0): 234 (2,3): 702 (2,6): 351 (3,0): 1053 (3,3): 2847 (3,6): 2496 (3,9): 1053
trapping sets H_Z (1,3)×117 (2,0)×234 (3,0)×1053 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 117 (1,6): 117 (2,0): 234 (2,3): 702 (2,6): 351 (3,0): 1053 (3,3): 2847 (3,6): 2496 (3,9): 1053

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C117 = SmallGroup(117, 2) (order 117), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 234; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 3, 9, 10, 19, 29}, b = {0, 3, 9} as element indices. Check weight 9. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.
model Mimo-V2.6-Flash (claimed, not verified)
reference arXiv:2610.13880
date 2026-10-12
notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[234,156,2]] — two-block group-algebra code over C117

Direction & hypothesis

Cell weight-9plus x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

  • X side: research/kit/surrogate.lightest_logical at 2,000 trials
  • Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

  • Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

    • connected Tanner graph and a connected stabilizer group.

  • Fitting a single-layer 2D layout to reach the local-2d cells: searched with

    • research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

  • Hypergraph products of two parents from this pool: connectivity passes, but n

    • multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 234

Parameters are also spelled out in provenance.construction of codes/234-156-2.json.

Parity checks

X-checks 117 (max weight 9) · Z-checks 117 (max weight 9)
H_X (117 checks, sparse supports)
[0, 3, 9, 102, 110, 115, 117, 120, 126] [1, 6, 14, 107, 113, 116, 118, 123, 131] [1, 2, 6, 8, 14, 17, 119, 125, 134] [0, 3, 9, 102, 110, 115, 117, 120, 126] [4, 11, 20, 96, 105, 112, 121, 128, 137] [4, 5, 11, 13, 20, 23, 122, 130, 140] [1, 6, 14, 107, 113, 116, 118, 123, 131] [5, 7, 13, 16, 23, 26, 124, 133, 143] [1, 2, 6, 8, 14, 17, 119, 125, 134] [0, 3, 9, 102, 110, 115, 117, 120, 126] [0, 3, 9, 10, 19, 29, 127, 136, 146] [4, 11, 20, 96, 105, 112, 121, 128, 137] [10, 12, 19, 22, 29, 32, 129, 139, 149] [4, 5, 11, 13, 20, 23, 122, 130, 140] [1, 6, 14, 107, 113, 116, 118, 123, 131] [12, 15, 22, 25, 32, 35, 132, 142, 152] [5, 7, 13, 16, 23, 26, 124, 133, 143] [1, 2, 6, 8, 14, 17, 119, 125, 134] [2, 8, 17, 18, 28, 38, 135, 145, 155] [0, 3, 9, 10, 19, 29, 127, 136, 146] [4, 11, 20, 96, 105, 112, 121, 128, 137] [18, 21, 28, 31, 38, 41, 138, 148, 158] [10, 12, 19, 22, 29, 32, 129, 139, 149] [4, 5, 11, 13, 20, 23, 122, 130, 140] [21, 24, 31, 34, 41, 44, 141, 151, 161] [12, 15, 22, 25, 32, 35, 132, 142, 152] [5, 7, 13, 16, 23, 26, 124, 133, 143] [7, 16, 26, 27, 37, 47, 144, 154, 164] [2, 8, 17, 18, 28, 38, 135, 145, 155] [0, 3, 9, 10, 19, 29, 127, 136, 146] [27, 30, 37, 40, 47, 50, 147, 157, 167] [18, 21, 28, 31, 38, 41, 138, 148, 158] [10, 12, 19, 22, 29, 32, 129, 139, 149] [30, 33, 40, 43, 50, 53, 150, 160, 170] [21, 24, 31, 34, 41, 44, 141, 151, 161] [12, 15, 22, 25, 32, 35, 132, 142, 152] [15, 25, 35, 36, 46, 56, 153, 163, 173] [7, 16, 26, 27, 37, 47, 144, 154, 164] [2, 8, 17, 18, 28, 38, 135, 145, 155] [36, 39, 46, 49, 56, 59, 156, 166, 176] [27, 30, 37, 40, 47, 50, 147, 157, 167] [18, 21, 28, 31, 38, 41, 138, 148, 158] [39, 42, 49, 52, 59, 62, 159, 169, 179] [30, 33, 40, 43, 50, 53, 150, 160, 170] [21, 24, 31, 34, 41, 44, 141, 151, 161] [24, 34, 44, 45, 55, 65, 162, 172, 182] [15, 25, 35, 36, 46, 56, 153, 163, 173] [7, 16, 26, 27, 37, 47, 144, 154, 164] [45, 48, 55, 58, 65, 68, 165, 175, 185] [36, 39, 46, 49, 56, 59, 156, 166, 176] [27, 30, 37, 40, 47, 50, 147, 157, 167] [48, 51, 58, 61, 68, 71, 168, 178, 188] [39, 42, 49, 52, 59, 62, 159, 169, 179] [30, 33, 40, 43, 50, 53, 150, 160, 170] [33, 43, 53, 54, 64, 74, 171, 181, 191] [24, 34, 44, 45, 55, 65, 162, 172, 182] [15, 25, 35, 36, 46, 56, 153, 163, 173] [54, 57, 64, 67, 74, 77, 174, 184, 194] [45, 48, 55, 58, 65, 68, 165, 175, 185] [36, 39, 46, 49, 56, 59, 156, 166, 176] [57, 60, 67, 70, 77, 80, 177, 187, 197] [48, 51, 58, 61, 68, 71, 168, 178, 188] [39, 42, 49, 52, 59, 62, 159, 169, 179] [42, 52, 62, 63, 73, 83, 180, 190, 200] [33, 43, 53, 54, 64, 74, 171, 181, 191] [24, 34, 44, 45, 55, 65, 162, 172, 182] [63, 66, 73, 76, 83, 86, 183, 193, 203] [54, 57, 64, 67, 74, 77, 174, 184, 194] [45, 48, 55, 58, 65, 68, 165, 175, 185] [66, 69, 76, 79, 86, 89, 186, 196, 206] [57, 60, 67, 70, 77, 80, 177, 187, 197] [48, 51, 58, 61, 68, 71, 168, 178, 188] [51, 61, 71, 72, 82, 92, 189, 199, 209] [42, 52, 62, 63, 73, 83, 180, 190, 200] [33, 43, 53, 54, 64, 74, 171, 181, 191] [72, 75, 82, 85, 92, 95, 192, 202, 212] [63, 66, 73, 76, 83, 86, 183, 193, 203] [54, 57, 64, 67, 74, 77, 174, 184, 194] [75, 78, 85, 88, 95, 98, 195, 205, 215] [66, 69, 76, 79, 86, 89, 186, 196, 206] [57, 60, 67, 70, 77, 80, 177, 187, 197] [60, 70, 80, 81, 91, 101, 198, 208, 218] [51, 61, 71, 72, 82, 92, 189, 199, 209] [42, 52, 62, 63, 73, 83, 180, 190, 200] [81, 84, 91, 94, 101, 104, 201, 211, 221] [72, 75, 82, 85, 92, 95, 192, 202, 212] [63, 66, 73, 76, 83, 86, 183, 193, 203] [84, 87, 94, 97, 104, 106, 204, 214, 223] [75, 78, 85, 88, 95, 98, 195, 205, 215] [66, 69, 76, 79, 86, 89, 186, 196, 206] [69, 79, 89, 90, 100, 109, 207, 217, 226] [60, 70, 80, 81, 91, 101, 198, 208, 218] [51, 61, 71, 72, 82, 92, 189, 199, 209] [90, 93, 100, 103, 109, 111, 210, 220, 228] [81, 84, 91, 94, 101, 104, 201, 211, 221] [72, 75, 82, 85, 92, 95, 192, 202, 212] [93, 96, 103, 105, 111, 112, 213, 222, 229] [84, 87, 94, 97, 104, 106, 204, 214, 223] [75, 78, 85, 88, 95, 98, 195, 205, 215] [78, 88, 98, 99, 108, 114, 216, 225, 231] [69, 79, 89, 90, 100, 109, 207, 217, 226] [60, 70, 80, 81, 91, 101, 198, 208, 218] [99, 102, 108, 110, 114, 115, 219, 227, 232] [90, 93, 100, 103, 109, 111, 210, 220, 228] [81, 84, 91, 94, 101, 104, 201, 211, 221] [93, 96, 103, 105, 111, 112, 213, 222, 229] [84, 87, 94, 97, 104, 106, 204, 214, 223] [87, 97, 106, 107, 113, 116, 224, 230, 233] [78, 88, 98, 99, 108, 114, 216, 225, 231] [69, 79, 89, 90, 100, 109, 207, 217, 226] [99, 102, 108, 110, 114, 115, 219, 227, 232] [90, 93, 100, 103, 109, 111, 210, 220, 228] [93, 96, 103, 105, 111, 112, 213, 222, 229] [87, 97, 106, 107, 113, 116, 224, 230, 233] [78, 88, 98, 99, 108, 114, 216, 225, 231] [99, 102, 108, 110, 114, 115, 219, 227, 232] [87, 97, 106, 107, 113, 116, 224, 230, 233]
H_Z (117 checks, sparse supports)
[0, 3, 9, 117, 120, 126, 127, 136, 146] [1, 6, 14, 118, 119, 123, 125, 131, 134] [2, 8, 17, 119, 125, 134, 135, 145, 155] [0, 3, 9, 117, 120, 126, 127, 136, 146] [4, 11, 20, 121, 122, 128, 130, 137, 140] [5, 13, 23, 122, 124, 130, 133, 140, 143] [1, 6, 14, 118, 119, 123, 125, 131, 134] [7, 16, 26, 124, 133, 143, 144, 154, 164] [2, 8, 17, 119, 125, 134, 135, 145, 155] [0, 3, 9, 117, 120, 126, 127, 136, 146] [10, 19, 29, 127, 129, 136, 139, 146, 149] [4, 11, 20, 121, 122, 128, 130, 137, 140] [12, 22, 32, 129, 132, 139, 142, 149, 152] [5, 13, 23, 122, 124, 130, 133, 140, 143] [1, 6, 14, 118, 119, 123, 125, 131, 134] [15, 25, 35, 132, 142, 152, 153, 163, 173] [7, 16, 26, 124, 133, 143, 144, 154, 164] [2, 8, 17, 119, 125, 134, 135, 145, 155] [18, 28, 38, 135, 138, 145, 148, 155, 158] [10, 19, 29, 127, 129, 136, 139, 146, 149] [4, 11, 20, 121, 122, 128, 130, 137, 140] [21, 31, 41, 138, 141, 148, 151, 158, 161] [12, 22, 32, 129, 132, 139, 142, 149, 152] [5, 13, 23, 122, 124, 130, 133, 140, 143] [24, 34, 44, 141, 151, 161, 162, 172, 182] [15, 25, 35, 132, 142, 152, 153, 163, 173] [7, 16, 26, 124, 133, 143, 144, 154, 164] [27, 37, 47, 144, 147, 154, 157, 164, 167] [18, 28, 38, 135, 138, 145, 148, 155, 158] [10, 19, 29, 127, 129, 136, 139, 146, 149] [30, 40, 50, 147, 150, 157, 160, 167, 170] [21, 31, 41, 138, 141, 148, 151, 158, 161] [12, 22, 32, 129, 132, 139, 142, 149, 152] [33, 43, 53, 150, 160, 170, 171, 181, 191] [24, 34, 44, 141, 151, 161, 162, 172, 182] [15, 25, 35, 132, 142, 152, 153, 163, 173] [36, 46, 56, 153, 156, 163, 166, 173, 176] [27, 37, 47, 144, 147, 154, 157, 164, 167] [18, 28, 38, 135, 138, 145, 148, 155, 158] [39, 49, 59, 156, 159, 166, 169, 176, 179] [30, 40, 50, 147, 150, 157, 160, 167, 170] [21, 31, 41, 138, 141, 148, 151, 158, 161] [42, 52, 62, 159, 169, 179, 180, 190, 200] [33, 43, 53, 150, 160, 170, 171, 181, 191] [24, 34, 44, 141, 151, 161, 162, 172, 182] [45, 55, 65, 162, 165, 172, 175, 182, 185] [36, 46, 56, 153, 156, 163, 166, 173, 176] [27, 37, 47, 144, 147, 154, 157, 164, 167] [48, 58, 68, 165, 168, 175, 178, 185, 188] [39, 49, 59, 156, 159, 166, 169, 176, 179] [30, 40, 50, 147, 150, 157, 160, 167, 170] [51, 61, 71, 168, 178, 188, 189, 199, 209] [42, 52, 62, 159, 169, 179, 180, 190, 200] [33, 43, 53, 150, 160, 170, 171, 181, 191] [54, 64, 74, 171, 174, 181, 184, 191, 194] [45, 55, 65, 162, 165, 172, 175, 182, 185] [36, 46, 56, 153, 156, 163, 166, 173, 176] [57, 67, 77, 174, 177, 184, 187, 194, 197] [48, 58, 68, 165, 168, 175, 178, 185, 188] [39, 49, 59, 156, 159, 166, 169, 176, 179] [60, 70, 80, 177, 187, 197, 198, 208, 218] [51, 61, 71, 168, 178, 188, 189, 199, 209] [42, 52, 62, 159, 169, 179, 180, 190, 200] [63, 73, 83, 180, 183, 190, 193, 200, 203] [54, 64, 74, 171, 174, 181, 184, 191, 194] [45, 55, 65, 162, 165, 172, 175, 182, 185] [66, 76, 86, 183, 186, 193, 196, 203, 206] [57, 67, 77, 174, 177, 184, 187, 194, 197] [48, 58, 68, 165, 168, 175, 178, 185, 188] [69, 79, 89, 186, 196, 206, 207, 217, 226] [60, 70, 80, 177, 187, 197, 198, 208, 218] [51, 61, 71, 168, 178, 188, 189, 199, 209] [72, 82, 92, 189, 192, 199, 202, 209, 212] [63, 73, 83, 180, 183, 190, 193, 200, 203] [54, 64, 74, 171, 174, 181, 184, 191, 194] [75, 85, 95, 192, 195, 202, 205, 212, 215] [66, 76, 86, 183, 186, 193, 196, 203, 206] [57, 67, 77, 174, 177, 184, 187, 194, 197] [78, 88, 98, 195, 205, 215, 216, 225, 231] [69, 79, 89, 186, 196, 206, 207, 217, 226] [60, 70, 80, 177, 187, 197, 198, 208, 218] [81, 91, 101, 198, 201, 208, 211, 218, 221] [72, 82, 92, 189, 192, 199, 202, 209, 212] [63, 73, 83, 180, 183, 190, 193, 200, 203] [84, 94, 104, 201, 204, 211, 214, 221, 223] [75, 85, 95, 192, 195, 202, 205, 212, 215] [66, 76, 86, 183, 186, 193, 196, 203, 206] [87, 97, 106, 204, 214, 223, 224, 230, 233] [78, 88, 98, 195, 205, 215, 216, 225, 231] [69, 79, 89, 186, 196, 206, 207, 217, 226] [90, 100, 109, 207, 210, 217, 220, 226, 228] [81, 91, 101, 198, 201, 208, 211, 218, 221] [72, 82, 92, 189, 192, 199, 202, 209, 212] [93, 103, 111, 210, 213, 220, 222, 228, 229] [84, 94, 104, 201, 204, 211, 214, 221, 223] [75, 85, 95, 192, 195, 202, 205, 212, 215] [96, 105, 112, 121, 128, 137, 213, 222, 229] [87, 97, 106, 204, 214, 223, 224, 230, 233] [78, 88, 98, 195, 205, 215, 216, 225, 231] [99, 108, 114, 216, 219, 225, 227, 231, 232] [90, 100, 109, 207, 210, 217, 220, 226, 228] [81, 91, 101, 198, 201, 208, 211, 218, 221] [102, 110, 115, 117, 120, 126, 219, 227, 232] [93, 103, 111, 210, 213, 220, 222, 228, 229] [84, 94, 104, 201, 204, 211, 214, 221, 223] [96, 105, 112, 121, 128, 137, 213, 222, 229] [87, 97, 106, 204, 214, 223, 224, 230, 233] [107, 113, 116, 118, 123, 131, 224, 230, 233] [99, 108, 114, 216, 219, 225, 227, 231, 232] [90, 100, 109, 207, 210, 217, 220, 226, 228] [102, 110, 115, 117, 120, 126, 219, 227, 232] [93, 103, 111, 210, 213, 220, 222, 228, 229] [96, 105, 112, 121, 128, 137, 213, 222, 229] [107, 113, 116, 118, 123, 131, 224, 230, 233] [99, 108, 114, 216, 219, 225, 227, 231, 232] [102, 110, 115, 117, 120, 126, 219, 227, 232] [107, 113, 116, 118, 123, 131, 224, 230, 233]
Code ID 234-156-2 · download JSON · raw on GitHub